Elliptic curve isogeny class 512.4-h over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.4-h
• Conductor: 512.4
• Rank: \( 0 \)

Base field \(\Q(\sqrt{17}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).

Elliptic curves in class 512.4-h over \(\Q(\sqrt{17}) \)

Isogeny class 512.4-h contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
512.4-h1	\( \bigl[0\) , \( 1\) , \( 0\) , \( -7 a + 19\) , \( -7 a + 19\bigr] \)
512.4-h2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 106 a - 257\) , \( 789 a - 2033\bigr] \)
512.4-h3	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a - 17\) , \( 9 a - 33\bigr] \)
512.4-h4	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -62 a - 97\) , \( 431 a + 673\bigr] \)
512.4-h5	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 19 a - 45\) , \( 972 a - 2491\bigr] \)
512.4-h6	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -982 a - 1537\) , \( 25287 a + 39489\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 8 & 4 \\ 8 & 1 & 2 & 4 & 4 & 8 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 2 & 4 & 2 & 1 & 4 & 2 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 8 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph